Library / Foundations: Physics / FPH-52
Foundations: Physics
Speed of a simple harmonic oscillator at a displacement
Speed of an undamped oscillator when it is a distance x from equilibrium.
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- FPH-52
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- Foundations: Physics
- Result
- |v| [m/s]
For an oscillator x = A cos ωt the speed is |v| = ω√(A² − x²), found by combining the energy balance with the position. It is greatest, ωA, at the centre (x = 0) and zero at the ends (x = ±A). The formula gives the size of the speed only; the direction depends on which way the body is moving. The displacement must not exceed the amplitude.
Speed of an undamped oscillator when it is a distance x from equilibrium. It is greatest at the centre and zero at the ends of the motion; the formula gives the size of the speed, not its direction.
Calculator
- Maximum speed, ωA (at the centre)0.5 m/s
- Fraction of the maximum speed at this displacement0.8
- Frequency, ω/2π1.592 Hz
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At 3 cm from the centre, with an amplitude of 5 cm and ω = 10 rad/s, the speed is 0.4 m/s, 80 % of the maximum 0.5 m/s.
- This is the 3–4–5 triangle: at 3 cm with a 5 cm amplitude, the speed is 4/5 of the maximum, a neat check.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| A Amplitude | 0.001 to 1 m | 0.1 | 9.995 | 9.895 | +10% | +15.2 % |
| ω Angular frequency | 1 to 100 rad/s | 0.04 | 4 | 3.96 | +10% | +10 % |
| x Displacement from equilibrium (not beyond the amplitude) | −1 to 1 m | — | — | — | +0.04 m | — |
Sweep of Angular frequency
| Angular frequency (rad/s) | Speed (m/s) |
|---|---|
| 1 | 0.04 |
| 13.38 | 0.535 |
| 25.75 | 1.03 |
| 38.13 | 1.525 |
| 50.5 | 2.02 |
| 62.88 | 2.515 |
| 75.25 | 3.01 |
| 87.63 | 3.505 |
| 100 | 4 |
Sweep one variable
Variables and how each one contributes
Speed
What the equation gives you. Everything else feeds it.
Angular frequency
The angular frequency of the oscillation, ω = 2πf, in rad/s. Raises the speed in direct proportion: a faster oscillation covers the same distance in less time. Raising it by 5 rad/s (to 15 rad/s) moves the result by +50 %, from 0.4 to 0.6 m/s.
Amplitude
The amplitude, the greatest displacement from equilibrium, in m. Raises the speed, strongly when x is close to A, because the body then moves just inside its range. Raising it by 0.01 m (to 0.06 m) moves the result by +29.9 %, from 0.4 to 0.5196 m/s.
Displacement from equilibrium (not beyond the amplitude)
The displacement from equilibrium at the moment considered, in m. Its size cannot exceed A. Lowers the speed as the body moves toward the end of its range, where it comes to rest. Raising it by 0.01 m (to 0.04 m) moves the result by −25 %, from 0.4 to 0.3 m/s.
Worked example with the values above
- Squares: A² = 0.0025 and x² = 9×10−4, so A² − x² = 0.0016 m².
- Square root: √0.0016 = 0.04 m.
- |v| = ω × 0.04 = 0.4 m/s.
Assumptions and limits
- An undamped simple harmonic oscillator: a restoring force proportional to the displacement.
- Constant amplitude and angular frequency; displacement is measured from the equilibrium position.
Common mistakes
- Entering a displacement larger than the amplitude.
- Using the frequency in hertz in place of the angular frequency in rad/s.
Statistics and uncertainty
The speed falls to zero as x approaches A, so the result is very sensitive to x near the ends: the same small error in x matters much more at 4.9 cm than at 1 cm (for A = 5 cm).
Where it is used
- The speed of a vibrating component at a given position.
- Checking velocity readings from a vibrating structure.
Maths you need
- Angular frequency: ω = 2π times the frequency, in rad/s
- Amplitude: the largest displacement from equilibrium